When the model is misspecified, however, the learning rate is less about convenience and more about necessity. [1]} showed that the generalized Bayes update (REF ) is fundamental from a decision-theoretic point of view. Moreover, they argue that the learning rate \(\eta \)  naturally emerges since, roughly, the parameter \(\theta ^\dagger \)  being estimated is defined by minimizing the expectation of a loss function \(\theta \mapsto \log p_\theta \) , and since that minimization problem is invariant to scalar multiples of the loss, the learning rate should appear in the posterior (REF ). In fact, the loss function interpretation makes their result much more general. In many cases, it is more natural to formulate the inference problem with a loss function rather than a statistical model. These are often referred to as Gibbs posterior distributions; see [2]}, [3]}, [4]}, [5]}, [6]}, and [7]}.
